Our goal in this section is to proof Theorem 2.5 below. It is a generalized fibre theorem for closures of quadratic modules.
Before dealing with it, we describe the setup that we will use, explain some constructions, and give some helpful results. So let A be an R-algebra. Let X be a Hausdorff topological space and
ˆ: A → C(X,R)
a morphism of R-algebras. Denote the image of A in C(X,R) by ˆA. As in Theorem 1.8 we assume condition (∗) to hold, i.e.
there exists p∈ A such that ˆp≥ 0 onX and for alli ∈ N, the set Xi = {x ∈ X | p(x)ˆ ≤ i} is compact. This in particular implies thatX is locally compact, as observed in [M1]. Note that in case X is compact, assumption (∗) is always fulfilled with p = 1, and if A is finitely generated by x1, . . . , xn and X ⊆ V(R) closed, we can choose p = x21 +· · ·+x2n.
Replacing pby p+ 1 if necessary, we can assume without loss of generality that ˆp is strictly positive on X. So 1pˆ is a continuous positive function on X, vanishing at infinity. This means that it takes arbitrary small values outside of compact sets.
Now we make the additional assumption that the functions ˆa which are bounded on X separate points. That means, for any two distinct points in X there is some a ∈ A, such that ˆa is bounded on X and takes different values in the two points. This is for example fulfilled if X is a compact subset of V(R), as elements from A separate points of V(R). However, there are also non-compact examples, as we will see later.
We use the assumption to apply [Bu], Theorem 3, which says that the bounded functions from ˆAlie dense in the set of bounded continuous functions on X, under the locally convex topology defined by the family of seminorms
||f||ψ := sup
x∈X|ψ(x)·f(x)|,
where ψ is a continuous function vanishing at infinity. We will use the seminorm defined by 1pˆ. Note that in case X is compact, we could use the standard Stone-Weierstrass approximation in-stead of [Bu] in the following proofs.
We begin with a technical lemma.
Lemma 2.3. Let ν be a positive regular Borel measure on X and h: X →R a measurable function. Suppose for all a ∈ A we characteristic function of Ai, so
Z
X
χhdν ≤ −1
iν(Ai) < 0.
Now choose a sequence (fn)n∈N of continuous functions on X with values in [0,1], that converges pointwise except on a ν-null set to χ. This can be done, using the regularity of ν as well as Urysohn’s Lemma as stated for example in [Ru]. Using [Bu], Theorem 3, we find a sequence (an)n∈N in A such that
on X. From this we get |ˆan| ≤ pˆ+ 1 on X, as √
by assumption, the Theorem of Majorized Convergence applies and yields is bounded from above by the function |h| which has a finite integral, we get in the same way as above
Z
Combining these result we have Z
which contradicts our assumption. So ν(Ai) = 0 for all i, which proves the result.
Towards the main theorem, we need a second R-algebra B and an algebra homomorphism ϕ: A →B. So we have the following diagram:
B
A b //
ϕ
OO
C(X,R)
Suppose M ⊆ B is a quadratic module. We want to describe the closure of M in terms of fibre-modules, indexed by elements from X. Namely, for x ∈ X, we denote by Ix the ideal in B generated by the set
{ϕ(a) | a ∈ A, ˆa(x) = 0}.
We call Mx := M + Ix the fibre-module to x, and we want to prove
M = \
x∈X
Mx.
For this we have to make more assumptions. Namely, suppose ϕ(a) ∈ M whenever ˆa ≥ 0 on X. This assumption is fulfilled in a large class of examples, as we will see below.
Now take L ∈ M∨, i.e. L is a linear functional on B that maps M to [0,∞). For b ∈ B we define a linear functional Lb on A by
Lb(a) := L(b·ϕ(a)).
We can apply Haviland’s Theorem (Theorem 1.8) to the func-tionals Lb2. Indeed, whenever ˆa ≥ 0 on X, then ϕ(a) ∈ M, so also b2 ·ϕ(a) ∈ M, so
Lb2(a) =L(b2 ·ϕ(a)) ≥ 0.
So we get positive regular Borel measures νb on X such that Lb2(a) = L(b2 ·ϕ(a)) =
Z
X
ˆ adνb
holds for alla ∈ A. As all considered measures are defined on X, we omit X under the integral sign from now on. The following result is a key ingredient for the proof of Theorem 2.5:
Proposition 2.4. For all b ∈ B νb ¿ν1,
that is, every ν1-null set is also a νb-null set.
Proof. Let b ∈ B be fixed and suppose N ⊆ X is a Borel set with ν1(N) = 0. We have to show νb(N) = 0. Denote the characteristic function of N by χ.
Choose a sequence of functions (fn)n∈N from C(X,[0,1]) that by the Theorem of Majorized Convergence.
Apply [Bu], Theorem 3, to obtain a sequence (an)n∈N from A so the Theorem of Majorized Convergence implies
¯¯
Combined with (2) we get Z
ˆ
a2ndν1 n→∞−→ 0. (3) Using the inequality from Lemma 1.4, we have
µZ
Together with (3) we find Z we get, again by Majorized Convergence,
¯¯ to χ, combined with (4) and (5), finally yields
0 = lim
n→∞
Z pfndνb = Z
χdνb = νb(N).
Proposition 2.4 allows us to apply the Radon-Nikodym
holds for all a ∈ A. Before stating and proving Theorem 2.5 below, we look at some properties of the functions θb.
For b1, b2 ∈ B, r1, r2 ∈ R and all a ∈ A we have and −h. The conditionR
ˆ so again by Lemma 2.3,
θm ≥ 0,
except on a ν1-null set that depends on m.
Last, for a, c∈ A and b ∈ B we have Z
ˆ
c·θb·ϕ(a)dν1 = Lb·ϕ(a)(c)
= L(b·ϕ(a)ϕ(c))
= Lb(ac)
= Z
ˆ
c·ˆaθb dν1.
So θb·ϕ(a) = ˆa·θb, except on a ν1-null set depending on a and b.
We are now prepared for the main theorem.
Theorem 2.5. Let A, B be R-algebras of countable vector space dimension, and let ϕ: A → B be an R-algebra homomorphism.
Let X be a Hausdorff space,
ˆ: A → C(X,R)
anR-algebra homomorphism fulfilling(∗) (see Theorem 1.8) and suppose the set
{ˆa | a ∈ A, ˆa bounded on X}
separates points of X. Further suppose M is a quadratic module in B and ϕ(a) ∈ M whenever aˆ≥ 0 on X. For x ∈ X denote by Ix the ideal in B generated by {ϕ(a) | a ∈ A, ˆa(x) = 0}. Then
M = \
x∈X
M + Ix.
Proof. One inclusion is obvious. For the other one fix q ∈ T
x∈X M +Ix and L ∈ M∨. We have to show L(q) ≥ 0. From L we construct all the functions θb as explained above.
Let B0 ⊆ B andA0 ⊆ A be a countable linear basis of B and A, respectively. Using the fact that each element inB is a difference of two squares, we can assume that B0 consist only of squares.
Denote the Q-subspace of A spanned by A0 by A. Let B be the Q-subalgebra of B generated by
B0 ∪ϕ(A0)∪ {q}.
B is a countable set and ϕ(A) ⊆ B. For each element b ∈ B we have a unique representation as a finite sum b = P
ri · bi, where all bi ∈ B0 and all ri ∈ R. Using the above demonstrated properties of the functions θb, we can find one single ν1-null set N ⊆ X such that for all x ∈ X \ N the following conditions hold:
(i) θb(x) =P
riθbi(x) for all b ∈ B (ii) θm(x) ≥0 for all m ∈ M ∩ B
(iii) θb·ϕ(a)(x) = ˆa(x)·θb(x) for all b ∈ B and all a ∈ A.
Because A and B are countable sets, this can indeed be ensured with one single null set N.
For x ∈ X \N we get linear functionals Lx on B by defining them on the basis B0:
Lx(b) := θb(x) for b∈ B0. For b∈ B with b = P
ri·bi as above we have Lx(b) =X
riLx(bi) = X
riθbi(x) =θb(x), where the last equality uses (i). So for m ∈ M ∩ B
Lx(m) = θm(x) ≥ 0
holds, using (ii). Now let b ∈ M be arbitrary, i.e. b is not necessarily from B. Write b = P
ri·bi with ri ∈ R and bi ∈ B0. As all bi are squares, b+P
ti·bi ∈ M whenever all ti ≥ 0. So b can be approximated in a finite dimensional R-subspace of B by
elements from M ∩ B. Just choose ti > 0 arbitrary small such that ri +ti ∈ Q. This shows
Lx(M) ⊆ [0,∞).
For a ∈ A and b∈ B we have b·ϕ(a) ∈ B and therefore Lx(b·ϕ(a)) = θb·ϕ(a)(x) = ˆa(x)·θb(x),
using (iii). Now take a ∈ A with ˆa(x) = 0. Approximate a by a sequence of elements an from A in the same way as above, so ˆ
an(x) n→∞−→ 0. So for b ∈ B, Lx(b·ϕ(a)) = lim
n→∞Lx(b·ϕ(an)) = lim
n→∞ˆan(x)θb(x) = 0.
By approximating arbitrary b ∈ B by elements of B we finally get
Lx(Ix) = {0}.
Defining
Lx ≡ 0 for x∈ N we have
Lx ∈ (M +Ix)∨ for all x ∈ X. (6) Now
L(q) = Lq(1) = Z
θq(x) dν1(x) = Z
Lx(q) dν1(x),
using the fact that q ∈ B and N is a ν1-null set. By (6) and our assumption on q, the function x 7→ Lx(q) is nonnegative on X, so L(q) ≥ 0.
In the following section, we show how an algebra A and a topo-logical spaceX can be constructed for a given quadratic module M in B, in a way that allows to deduce Schm¨udgen’s Theorem from Theorem 2.5.